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Homogeneity, Saddle Path Stability, and Logarithmic Preferences in Economic Models

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  • Dirk Bethmann

    ()
    (Department of Economics, Korea University)

Abstract

In a stylized Robinson Crusoe economy, we demonstrate the usefulness of homogeneity in initial conditions when solving and analyzing macroeconomic models. In a first step, we define state-like and control-like variables. In a second step, we introduce the value-function-like function. While the former step reduces the number of variables that have to be considered when solving the model, the latter step reduces the dimensionality of the Bellman equation associated with the optimization problem. The model’s solution is shown to be saddle-path stable, such that the phase diagram associated with the Bellman equation has two solution branches and the structure of our model allows us to state both the stable and the unstable branch explicitly. We also explain the usefulness of logarithmic preferences when studying the continuoustime Hamilton-Jacobi-Bellman equation. In this case the utility maximization problem can be transformed into an initial value problem for an ordinary differential equation.

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Bibliographic Info

Paper provided by Institute of Economic Research, Korea University in its series Discussion Paper Series with number 0702.

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Length: 20 pages
Date of creation: 2007
Date of revision:
Handle: RePEc:iek:wpaper:0702

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Keywords: Closed-form solution; saddle path; homogeneity in initial conditions; continuous time;

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  1. Ladron-de-Guevara, Antonio & Ortigueira, Salvador & Santos, Manuel S., 1997. "Equilibrium dynamics in two-sector models of endogenous growth," Journal of Economic Dynamics and Control, Elsevier, vol. 21(1), pages 115-143, January.
  2. Bond, Eric W. & Wang, Ping & Yip, Chong K., 1996. "A General Two-Sector Model of Endogenous Growth with Human and Physical Capital: Balanced Growth and Transitional Dynamics," Journal of Economic Theory, Elsevier, vol. 68(1), pages 149-173, January.
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  4. Abba P. Lerner, 1959. "Consumption-Loan Interest and Money: Rejoinder," Journal of Political Economy, University of Chicago Press, vol. 67, pages 523.
  5. Casey B. Mulligan & Xavier Sala-i-Martin, 1992. "Transitional Dynamics in Two-Sector Models of Endogenous Growth," NBER Working Papers 3986, National Bureau of Economic Research, Inc.
  6. Danyang Xie, 2002. "Divergence in Economic Performance: Transitional Dynamics with Multiple Equilibria," GE, Growth, Math methods 0210002, EconWPA.
  7. Bethmann, Dirk, 2008. "The open-loop solution of the Uzawa-Lucas model of endogenous growth with N agents," Journal of Macroeconomics, Elsevier, vol. 30(1), pages 396-414, March.
  8. Antonio Ladron de Guevara & Salvador Ortigueira & Manuel Santos, 1995. "A Two-Sector Model of Endogenous Growth with Leisure," Working Papers 9503, Centro de Investigacion Economica, ITAM.
  9. Ortigueira, Salvador & Santos, Manuel S., 2002. "Equilibrium Dynamics in a Two-Sector Model with Taxes," Journal of Economic Theory, Elsevier, vol. 105(1), pages 99-119, July.
  10. Caballe, Jordi & Santos, Manuel S, 1993. "On Endogenous Growth with Physical and Human Capital," Journal of Political Economy, University of Chicago Press, vol. 101(6), pages 1042-67, December.
  11. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
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